7 PRIMITIVE OF AN ANALYTIC FUNCTION 221

7. PRIMITIVE OF AN ANALYTIC FUNCTION IN A SIMPLY CONNECTED
DOMAIN

A simply connected domain A c C is an open connected set such that
any loop in A is homotopic in A to a loop reduced to a point', it is clear
that any open subset of C homeornorphic to A is a simply connected domain.

Example

(9.7.1) A star-shaped domain Ac C with respect to a point aeA is an
open set such that for any z e A, the segment joining a and z is contained
in A. Such a set is clearly connected ((3.19.1) and (3.19.3)); if y is any loop
in A, write (p(t, £) = a 4- (1 - £)(y(0 - *) for 0 < £ ^ 1; <p is a loop homotopy
of y into the loop reduced to a. An open ball is a star-shaped domain with
respect to any of its points.

(9.7.2) If A c C is an open connected set, for any two points u, v of A there
is a road of origin u and extremity v.

We need only prove that the subset B <r A of all extremities of roads
in A having origin u is both closed and open in A (Section 3.19), If x e A n B,
there is a ball S of center x contained in A, and by assumption S contains
the extremity v of a road y of origin u; the segment of extremities v, x is
contained in S, and if y is defined in [a, b], the road yt equal to y in [a, b]9
to y1(0 = v -f (t — b)(x — i?) in [b, b -f 1] is in A and has origin u, extremity x;
hence x e B. On the other hand, if .y e B, there is a ball S of center y contained
in A; for any t; e S, the segment of extremities y, v is contained in S and we
define in the same manner a road of origin u, extremity v9 which is in A, hence
S <= B. Q.E.D.

(9.7.3) If A c C is a simply connected domain, any function f analytic in A
has a primitive which is analytic in A.

Let a, z be two points of A, yi9 y2 two roads in A of origin a and extremity

z: then f f(x)dx= f f(x) dx. Indeed, we may suppose, by replacing y2

Jyi Jy2

by an equivalent road, that yl is defined in [b, c] and y2 in [c, a]; then

y = y± v yl is a circuit in A, which is therefore homotopic to a point in A,\^e, |£ - «'| < e imply \q>(t, 0 - (?('', Ol < P/4. Let (Oo««r